Multipeak Solutions For Critical Polyharmonic Equations

Polyharmonic equations with critical growth have a deep background in physics and geometry research.Due to the lack of compactness,singularity,degeneracy and some broken symmetries,it is very challenging to study this type of equations or systems.Therefore,the research on these problems has important theoretical significance and application value.In this thesis,we are concerned with the existence and multiplicity of multipeak solutions for some critical polyharmonic equations by combining finite-dimensional reduction method with minimax argument,energy expansion and local Pohozaev identities.Firstly,we consider the high-order Ambrosetti-Prodi type problem.To solve this problem,we construct two types of multipeak solutions,one is the peak solution with the concentrated point inside the domain,and the other is the peak solution with the concentrated point near the boundary of the domain.In order to construct these two types of solutions,we first reduce the problem into a finite-dimensional problem through finitedimensional reduction method.Then,two types of multipeak solutions with concentrated points inside the domain and near the boundary of the domain are obtained by energy expansion and minimax argument.Secondly,we consider the higher-order Henon type problem.We take the small disturbance ε of the critical exponent as the parameter,and the number of peaks k depends on the disturbance parameter ε.The problem is reduced into a finite-dimensional problem through finite-dimensional reduction method,and then the existence of the multipeak solution is obtained by energy expansion and direct elimination of Lagrange multiplier terms.Furthermore,we consider the high-order Brezis-Nirenberg type problem.We take an arbitrarily large positive integer k as a parameter and construct a k peak solution whose peaks evenly distribute on the circumference of a two-dimensional space by combining finite-dimensional reduction method with energy expansion and local Pohozaev identities.Since k can be chosen arbitrarily large,it shows that this problem has infinitely many solutions.Finally,we consider the high-order prescribed curvature problem.We take an arbitrarily large positive integer k as a parameter and obtain the existence of k peak solution concentrated near the stable critical point of a certain function by finite-dimensional reduction method,energy expansion and local Pohozaev identities.Further,the arbitrarily large selection of parameter k can imply the existence of infinitely many solutions.